Using Challenge Based Learning to Teach the
Fundamentals of Exponential Equations
Dr. Jeff Kastner, Dr. Joni Torsella
Department of Engineering Education
Dr. Anant R. Kukreti
Department of Biomedical, Chemical, and Environmental Engineering
University of Cincinnati
Cincinnati, OH 45221
Corresponding Email: [email protected]
Abstract
Our work is based on an effort to teach the concepts of exponential growth and decay functions
using the Challenge Based Learning approach (CBL). The cohort consisted of high-school and
middle-school science and math teachers. A unique aspect of this work was that in parallel to
teaching the mathematical concepts, the instructors were also explaining how to implement the
CBL pedagogy into a middle or high-school classroom. Thus the outcome of the project was a
class module that the teachers could take back to their school. The feedback from the cohort was
positive with regards to the mathematical concepts and the CBL pedagogy. Two main
improvements suggested were to separate the class in two based on mathematical level and to
help develop cost efficient activities which could be used in the middle and high schools. Both
of these suggestions are currently being implemented into the 2014 classes.
Introduction
Challenge Based Learning (CBL) is an inductive pedagogy which allows students to identify a
real-life problem that interests them and then learn what they need to know in order to solve the
problem.1-3 Information gained from doing assignments and projects can result in long term
retention of the information and make the student feel like their school work has purpose.
Mathematics classes work well with CBL since the equations being taught are often universal to
STEM disciplines. This allows for a large variety of topics to be covered under the umbrella of a
mathematical learning objective.
A flow chart of the components for CBL is provided in Figure 1. The pedagogy begins by
stating a Big Idea which ultimately will be linked to an academic standard or learning objective.
Based on the Big Idea, students begin to formulate a set of Essential Questions. The Essential
Questions are still very broad and just show that there are some important components of the Big
Idea which are not fully understood. One of the Essential Questions is then selected and
formulated into a Challenge which can be directly linked to a class unit. This is usually where
the students are asked to build, design, test, and/or model a real world problem. To begin
addressing the Challenge, the students are asked to form a set of Guiding Questions which will
ultimately lead to specific units and activities to be performed in the classroom to help answer
Proceedings of the 2014 ASEE North Central Section Conference
Copyright © 2014, American Society for Engineering Education
these Guiding Questions. The instructor works as a facilitator throughout this project and has to
make sure the Challenge and Guiding Questions remain in line with the course unit and/or
learning objectives.
Figure 1. Flow Chart for implementing Challenge Based Learning4
Course Description
Our work is based on an effort to teach the concepts of exponential growth and decay functions
using the CBL approach. The project was implemented in an Engineering Applications in Math
class consisting of middle and high-school teachers who had applied to a summer math, science
and engineering training program at our University.4 The objectives of the particular class were
to get middle-school and high-school teachers excited about math and engineering, to help
teachers better understand the link between engineering and math, and to give teachers the
resources they need to use similar math and engineering concepts in their classroom. The
varying backgrounds of the middle and high-school teachers led to a vast range of exponential
equations being applied in the CBL project. For the rest of this extended abstract, the teachers
were the students of the class, so to avoid confusion; they will be referred to as the cohort.
The course met 5 days per week for 5 weeks during July 2013. Monday, Wednesday and Friday
were one hour lectures and Tuesday and Thursday consisted of 2½ hour lab class. Typically
Monday would introduce the fundamental Math concept to be used during the week, Wednesday
would be additional content and sample problems, and then Friday would consist of an in-class
quiz and final synopsis of the week’s activities. The labs were structured to typically use both
the Tuesday and Thursday lab time to complete. While completing the labs, the cohort was
asked to answer questions about their observations and turn in their data sheet at the end of each
class with sample calculations included. Most lab activities took just under 2 hours to complete,
so there was plenty of in class time to complete the lab datasheets as a group.
Proceedings of the 2014 ASEE North Central Section Conference
Copyright © 2014, American Society for Engineering Education
The primary topics during lecture included: applications related to lines, linear functions,
exponential functions, trigonometry, and calculus. Each topic was covered for a week except
calculus which was only one lecture class and had no quiz. The lab classes which complemented
each topic included calibrating a linear force sensor, a structures lab, and a circuit lab. The
structures lab required the students to build a bridge and study the forces on the bridge. During
this lab they were asked to link the forces in the structure to the vector and trigonometric
equations presented during lecture. The second portion of the lab required each group to
measure the velocity of the car driving over the bridge. The circuit lab was covered over two lab
periods and required each group to build circuits with resistors in series and parallel. The circuit
lab finished with each group building and testing an RC circuit. The RC circuit portion provided
an introduction into exponential growth and decay functions.
Challenge Based Learning Project
After completing the circuit lab, the groups were asked to design a circuit which would model a
real-world problem that exhibited an exponential growth or decay phenomenon. This became
the Stated Challenge for the CBL project. There was a wide variety of real-world problems
selected including the loss of water in lakes near deserts, the tilt angle of the leaning tower of
Pisa, the decay rates of medication in the body, the population control of invasive species, etc.
All of these topics were self-selected by groups of two to three and all topics had to have an
exponential equation used in describing the problem. With the Stated Challenge, the students
then began to ask guiding questions. A few common Guiding Questions included:
“How does one determine the Time Constant/Rate of Decay/Growth from a given figure or set of
data?”
“How do you build a circuit which models a real-world problem?”
“How do you use the Circuit and Mathematical Model to help solve the real-world problem?”
“How do you generate a similar in-class activity for middle and high school students?”
The instructors of the class then guided the students to see how exponential functions are linked
to their Global Problem and how solutions to their problems were linked to engineering fields.
Two lessons resulted from their stated challenges: one on solving and plotting exponential
equations and one on how to take a given problem and model it with a real circuit. A circuit
design lab activity was linked to these two lessons using the circuit kits to model their real-world
problem and see their equations at work. To design the circuit, the students were required to go
through the Engineering Design Process and document the steps taken to complete the design.
Also, after the first lab class, the students were given a second lab class to re-design their circuit
or adjust their model. This second portion of the project helped ensure that everyone had a
working circuit at the end of the class.
The final stage of the project required the students to present their design and results to their
colleagues. The presentations required each group to describe how their project could become
its own CBL project. It was during this portion of the class that many others in the classroom
began to see the power of a single math equation and how many different topics all seemed to be
using the same equation. Also, making the groups present their work allowed them to take
Proceedings of the 2014 ASEE North Central Section Conference
Copyright © 2014, American Society for Engineering Education
possession of the project and take the final steps to showing that they understood what they were
presenting. Each group was only given 7 minutes, and based on the importance and positive
feedback from this portion of the class, it has been decided that each group will give a longer
presentation for the next run of the class.
Course Feedback
Evaluations were given at the end of the course which included questions on a 5-point Likert
scale and a comment section. The two most positive 5-point scale questions were that the
students felt the course helped them understand how math and science knowledge leads to
different STEM career choices, and they felt the activities and projects helped cultivate effective
team work. This shows that the most positive feedback on the course was related to the CBL
project and the ability to do activities. For the upcoming year, this will be further emphasized by
increasing the activity time.
The weakest point total for the survey was the students didn’t feel that the instructors presented
the concepts effectively, and the projects in the class could not be applied in their classroom.
This again was great feedback and the second item will be addressed during this summer by
making the teachers aware of cost-effective methods to do the same activities. The issue with the
course concepts was highlighted by the large standard deviation for answers to this question and
was further highlighted by the comments. One teacher said: “I learned new math concepts that I
didn’t know before” while another teacher stated that “the only thing this course did for me was
enhance the topics I have already been certified to teach.” This breadth in response for a
mathematics course makes it difficult for teaching concepts since the cohort had such a wide
range of background. During the upcoming year, the plan is to split the class so there is a section
for high-school math teachers and a section for all science teachers and middle school math
teachers. Ultimately, some of the high school science teachers could take the more advanced
topics class.
Some comments from the students directly related to the CBL portion of the class include:
“I enjoyed connecting the math to the labs especially the circuits labs to see the math in action”
“I liked to see how the math connected to engineering careers.”
“I really enjoyed the projects”
“Modeling Project – the pay-off was great and I enjoyed seeing all the other presentations”
“I would enjoy more time to discuss lab results”
The instructors of the course had a very positive view of the CBL portion of the class. This
included more questions from the students during lecture because they were seeing how the
lecture could help their project. This really showed that they were taking possession of the
material. The students also were able to better express things they were confused about which
could easily be over looked during a traditional lecture format. The best example of this was
some projects were looking at time constants while some were looking at rates. To the
instructors they were just inverses of each other, but the students had a hard time seeing this and
Proceedings of the 2014 ASEE North Central Section Conference
Copyright © 2014, American Society for Engineering Education
requested more information presented to them during lecture. Finally, a few students went
beyond the expectation and brought in other equations to better match their data. This included
the “S” shaped logarithmic equation for population growth and saturation.
Conclusion
A Challenge Based Learning class module was introduced in a summer teacher training program.
The course itself was structured under the umbrella of CBL and at the same time the students had
to use the Engineering Design Process to design a circuit which models a real-world problem.
Overall, the student learning objectives were met from this project and it is believed the students
will have long term retention of the information. At the same time, the vast range of
mathematical backgrounds made it difficult for everyone to benefit from the class. This will be
addressed during the up-coming summer class where the class will be split into two separate
classes based on the individual’s mathematical background.
Acknowledgements
The author would like to acknowledge the financial support provided by the U.S. National
Science Foundation Award, DUE-1102990. Any opinions, findings, conclusions, and/or
recommendations are those of the investigators and do not necessarily reflect the views of the
Foundation.
(1) Apple
Inc.,
Challenge
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White
Paper,
October
2011
https://www.challengebasedlearning.org/public/admin/docs/CBL_Paper_October_2011.pdf
(2) Apple Inc., Challenge Based Learning, A classroom guide, January 2011.
https://www.apple.com/euro/education/docs/CBL_Classroom_Guide_Jan_2011.pdf
(3) Johnson, L.F., Smith, R.S., Smythe, J.T., Varon, R.K., “Challenge-Based Learning: An Approach for Our
Time,” Austin, Texas: The New Media Consortium, 2009.
(4) Kukreti, A.R., Rutz, E., Steimle, J., Jackson, H.E., Maltbie, C., “Training Secondary Math and Science
Teachers to Bring an Engineering Perspective to the Classroom,” Paper ID #8215, 120 th ASEE Annual
Conference & Exposition, June 2013.
Proceedings of the 2014 ASEE North Central Section Conference
Copyright © 2014, American Society for Engineering Education